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Both spin angular momentum (SAM) and orbital angular momentum (OAM) can be used to carry information in classical optics and quantum optics. In this paper, the encoding of angular momentum (AM) information of photons onto surface plasmon polaritons (SPPs) is demonstrated using a nano-ring plamonic lens. Near-field energy distribution on the metal surface is measured using a near-field scanning optical microscope (NSOM) when the plasmonic lens is excited by photons with different combinations of SAM and OAM. It is found that both the SAM and OAM can influence the near field energy distribution of SPPs. More interestingly, numerical and experimental studies reveal that the energy distribution on the plasmonic lens surface is determined by the absolute value of the total AM. This gives direct evidences that SPPs can be encoded with the photonic SAM and OAM information simultaneously and the spin degeneracy of the photons can be removed using the interactions between photonic OAM and plasmonic lens. The findings are useful not only for the fundamental understanding of the photonic AM but also for the future design of plasmonic quantum optics devices and systems.
©2012 Optical Society of America
1. Introduction
Photonics plays an important role in the modern information era due to the high information capacity and fast propagation speed of light. Information can be encoded in many properties of photons, including the angular momentum (AM) that can be divided into spin angular momentum (SAM) and orbital angular momentum (OAM) [1] SAM is associated with the polarization of light and has been widely explored in different optical applications; while OAM arises from the spatial distribution of the intensity and the azimuthal phase of an optical field. Compared with SAM, OAM represents a fundamentally new optical degree of freedom and attracted increasing attention in recent years. Unlike the commonly used fundamental Gaussian beam, photons carrying OAM have a doughnut shape spatial distribution and a helical azimuthal phase. Laser beams with OAM have found applications in trapping and rotating particles and living cells [2–4], manipulating individual spins with nanoscale resolution in diamond [5], superhigh-density optical data storage, imaging [6], metrology [7–10], and free space communications [11]. Potential applications of photons with OAM in the quantum information field to supply a platform for the investigation of high-dimensional entanglement have also been explored [12–15].
With the rapid progress of nano-fabrication technology, information carried by photons can be transferred or encoded through the interactions between photons and nano-scale structures. Surface plasmon polaritons (SPPs), collective oscillating electrons excited by electromagnetic field, offers a particular attractive means due to its capabilities to confine the electromagnetic energy at a nanoscale volume far beyond the diffraction limit in the visible and near infrared spectrum regime. Unique properties of SPPs have found a wide range of applications [16, 17], including nanoscale optical waveguiding, perfect lensing, extraordinary optical transmission, subwavelength lithography, and ultrahigh sensitivity biosensing. Among them, transferring SAM from photons to SPPs have been studied using different types of nano-structures, including nanohole arrays [18–20], nano-scale waveguides [21–25], plasmonic nano-lens [26–29], and so on. Recently, encoding OAM information on SPPs in nano-structures was also discussed with similar structures [28, 30–34]. Further studies proved that SPPs were useful in quantum information through preserving the SAM entanglement or OAM entanglement in plasmonic devices [35–38].
In this paper, we study the excitation of SPPs in a nano-ring structure using photons with different combinations of SAM and OAM. We demonstrate that different combinations of the SAM and OAM lead to different intensity distributions of the SPPs via nanostructures [39]. For photons with spin only, the near field SPP energy distributions are indistinguishable and exhibit spin degeneracy. On the contrary, the SPP energy distributions excited by photons with combination of SAM and OAM are primarily determined by the absolute value of total angular momentum. The spin degeneracy can be removed through the interactions between photonic OAM and plasmonic lens. The theoretically infinite number of OAM eigenstates offers a large capacity for encoding the SPPs to carry information. These findings may find broad applications in the future integrated plasmonic and quantum optic circuits for quantum information processing and encryption.
2. Experimental setup and methods
The experimental setup is illustrated in Fig. 1 . A laser beam (671 nm wavelength) from a semiconductor laser diode (LD) is focused onto a nano-ring plasmonic lens sample using an objective lens (10X, NA = 0.25). Computer-generated holograms (CGHs) are used to manipulate the OAM of the incident light, and the SAM can be adjusted by the combination of a polarizer and a quarter wave plate. A scanning electron microscope (SEM) image of the nano-ring plasmonic lens is shown as the inset (2) in Fig. 1. The nano-ring is fabricated by focused ion beam (FIB) milling in a layer of 135 nm thick gold film evaporated on a 1cm × 1cm glass substrate. The ring has a diameter of 6 µm. The width and depth of the groove are 200 nm and 75 nm respectively. The sample is placed on a 2D piezoelectric stage to facilitate the alignment of the plasmonic lens and the laser beam. A collection mode near-field scanning optical microscope (NSOM, Nanonics MultiView 4000TM) with 100 nm diameter aperture probe is used to scan the sample at approximately 200 nm above the sample. The collected signal is guided to an avalanche photodiode (APD) through a polarization maintaining single mode fiber. The MultiView 4000TM system is used to control the movement of the NSOM probe and form NOSM images.
Fig. 1 Schematic of the experimental setup. A laser beam of 671 nm wavelength from a laser diode (LD) illuminates the sample at normal incidence to excite the SPPs. A computer-generated hologram (CGH) is used to control the OAM states and a combination of a PBS and a λ/4 wave plate is used to modulate the SAM states of excitation photons. NSOM images are recorded by a Nanonics MultiView 4000TM system. Inset (1) is the 3D drawing of the nano-ring structure excited by photons with different SAM and OAM. An SEM picture of the nano-ring structure fabricated in a gold film is shown in Inset (2). Inset (3) is the diagram of a single ring plasmonic lens and the coordinates used in calculation. The illumination is along the z direction.
The proposed structure and the coordinates for calculation are illustrated in the inset (3) of Fig. 1. A single ring slot is etched into a thin metal film deposited on glass substrate. Light illuminates the structure normally from the substrate side. Considering an incident light with both OAM of l (l = 0, 1, 2…) and SAM of σ (σ = −1, 1), it can be expressed in cylindrical coordinates as
where eiσφ is the Berry phase due to the SAM and eilφ is the phase due to the OAM. For a sufficiently narrow slot, only the radial component can couple to excite SPPs. Thus the plasmonic field at an observation point (R, θ) near the origin due to the excitation along an incremental length of the annular slot is given bywhere E0z is a constant that is related to the coupling efficiency, r is the radius of the nano-ring and is the wave vector of the SPPs that propagate from the excitation location to the center, is the wave vector of the SPPs that propagate in the z-direction. Here we used the fact that the SPPs are substantially polarized in the z-direction (normal to the surface). The subwavelength slot opening can be regarded as an array of secondary sources. The plasmonic field at the observation point can be expressed asFor simplicity, we neglect the propagation loss of the SPPs, i.e. Im(kr) = 0, and then. With these assumptions Eq. (3) can be simplified as:where J (l+σ) is the (l + σ)th order Bessel function of the first kind [32]. This simple analytical derivation clearly indicates that the plasmonic field near the center of the nano-ring is determined by the total AM.It is worth mentioning that the coupling efficiency of longitudinal component into the aperture NSOM probe is much lower than the transversal components for extremely small aperture size [26, 40]. The expected NSOM signal dominated by the transversal component contribution can be shown to be proportional to . According to Eq. (4), the expected NSOM signal should be proportional to the square of the differentiation of the Bessel function as . For example, for the case of l + σ = 1, the longitudinal electric field shows a profile of 1th order Bessel function which is a doughnut shape with a dark center, whereas the expected NSOM signal is a combination of 0th and 2th order Bessel functions, which leads to profiles of a solid spot in the center and a donut shape surrounding it, respectively.
Photons with different combinations of OAM and SAM (as listed in Table 1 ) are used to excite the SPPs on the sample. In Table 1, l and σ represent the OAM and SAM of the photons respectively (the unit of ħ is ignored for simplification), where σ = −1 corresponds to the right-handed circular polarization and σ = + 1 corresponds to the left-handed circular polarization, and j = l + σ represents the total AM of the photon. The analytical expression derived above only considers the z-component of the SPP field and ignored the finite width of the nano-ring. In this work, numerical results are also calculated with finite element method (COMSOL Multiphysics 4.1) models to more precisely predict the theoretical SPP field near the center. The parameters of the structure are the same as those used in the experiment. The index of refraction for gold is chosen to be 0.163 + 3.4633i at the 671 nm illumination wavelength [41], and the index of refraction for glass substrate is chosen to be 1.5. Both energy distributions for the longitudinal and transversal components at 200 nm above the sample are calculated for various combinations of SAM and OAM listed in Table 1.
Table 1. The Matrix of Total AM with Different Combinations of OAM and SAM.
3. Results and discussions
We first consider only the effect of the SAM on the distribution of the SPPs. This is realized by using a fundamental Gaussian (l = 0) to excite the SPPs. The SPPs excited at different points of the nano-ring propagate towards and interfere at the center of the ring and form specific patterns of intensity distribution [42]. The experimental and numerical results are summarized in Fig. 2 , where σ of the excitation light on the upper and lower rows are 1 and −1 respectively. The experimental results are shown in the first column, and the second and third columns are the numerical simulations of the transversal and the longitudinal electric field distributions of the SPPs, respectively. In our experiment, only the transversal electric fields of the SPPs are detected by the NSOM due to the fact that the coupling efficiency of the transverse component is much higher than the longitudinal component. In both cases (Fig. 2(a) and 2(d)), the transversal SPPs interfere constructively at the center of the nano-ring and form a bright spot. The values of full with at half maximums (FWHMs) of the center bright spots are 188nm in Fig. 2(a) and 185 nm in Fig. 2(d), agreeing well with the numerical simulations (Fig. 2(b) and 2(e)) with a FWHM of 184nm. It should be noted that the experimental results are not rotation symmetric as previous work [28]. The reason is that the collection probe used in the experiment is bent and thus has different collection efficiencies in different directions, which will be further discussed later. There is a dark center spot for the longitudinal component as shown in Fig. 2(c) and 2(f), which is due to a geometric π-phase difference for the counter-propagating SPPs excited at the opposite side of the nano-ring [43]. From both the numerical and experimental results, it can be clearly seen that the spatial distributions of the SPPs under σ = −1 and σ = 1 excitation are indistinguishable. This represents a spin degeneracy of the focusing properties of the nano-ring plasmonic lens.
Fig. 2 The intensity distributions of the SPPs excited by photons with different SAM, σ = 1 in a, b and c and σ = −1 in d, e and f. a and d are the experimental NSOM images, b and e are the numerical simulations of the transversal SPPs and c and f are the numerical simulations of the longitudinal SPPs. The color bars are shown to the right corner that is also used for the corresponding results shown in the following part of this work.
The phenomenon is quite different when we introduce the OAM of photons. The initial phase of 2lπ around the nano-ring, for optical excitation that carries OAM with a topological charge of l, leads to different SPP distributions near the center of the plasmonic lens for excitation photons with different SAM states. An example for the case of l = 2 is shown in Fig. 3 . The NSOM images of the SPPs with different SAM are now quite different from the cases shown in Fig. 2. Near the center of the plasmonic lens, the SPPs interfere destructively for σ = 1 to form a bright ring and constructively for σ = −1 to form a bright center spot, which is confirmed by the corresponding numerical results given in Fig. 2(c) for σ = 1 and Fig. 2(d) for σ = −1. The SPPs excited by photons with different SAM are distinguishable by introducing an OAM of l≠0. The removal of degeneracy between the SPPs with the SAM of σ = 1 and σ = −1 makes it possible for SPPs to carry information encoded on the SAM.
Fig. 3 The intensity distributions of the SPPs excited by photons with OAM of l = 2 and SAM of σ = ± 1. a and b are the NSOM images of the SPPs with SAM of σ = 1 and σ = −1 respectively. c and d are the numerical simulations of the transversal SPPs corresponding to a and b.
Through introducing the OAM, the capacity for encoding information in SPPs are also increased dramatically. Figure 4 summarizes the numerical and experimental results of the transverse field distributions for the SPPs excited by photons with different OAM (l = −1, 0, + 1 and + 2) and the same SAM (σ = 1). The intensity distributions near the center are different with different OAM, a ring with diameter (D) = 378 nm, a bright spot with FWHM = 188 nm, a bipartition ring with D = 318 nm and a bigger ring with D = 534nm in Fig. 4(a), 4(b), 4(c) and 4(d), respectively. The corresponding numerical results are given in Fig. 4(f), 4(g), 4(h) and 4(i) with corresponding values of 374 nm, 184 nm, 314 nm and 532 nm. In Fig. 4(c), the center ring appears to split into two parts and disagrees with the numerical result in Fig. 4(g). In fact, the broken of symmetry results from the asymmetric collection efficiency of the bent NSOM probe but it does not deter from revealing the intensity distribution of the SPPs on the plasmonic lens. For example, the SPPs intensity distribution along dotted line 1 (black) and 2 (green) in Fig. 4(e) (the same as Fig. 4(c)), are shown in Fig. 4(j) and their topography agrees well with each other, despite there is large absolute intensity difference between them. The one-to-one correspondence between the SPPs intensity distribution and the OAM makes the OAM states possible candidates for information encoding onto the SPPs. Although only four OAM states are presented here, the actual number of OAM eigenstates is infinite, which is much more than the number of photonics SAM eigenstates (only 2 eigenstates). This demonstrates the potential of using OAM states to realize super capacity information encoding on SPPs.
Fig. 4 The intensity distributions of the SPPs excited by photons with the same SAM (σ = 1) and different OAM. a, b, c and d are the SPPs carrying with OAM of l = - −1, 0, 1 and 2, with corresponding numerical simulations of the transversal SPPs in f, g, h and j, respectively. Line 1(black) and line 2(green) in Fig. 4 j are the topography of the SPPs intensity distribution along dotted line1 and 2 in Fig. 4e, which is the same with Fig. 4c.
The OAM is superior to the SAM in information encoding for more available states but not an arbitrary number of OAM states is available for information encoding. The following study will give the principle in using OAM for information encoding. We will demonstrate that the OAM and SAM play an equivalent role in determining the intensity distribution of the SPPs generated by the nano-ring plasmonic lens. The intensity distribution is mainly decided by the absolute value of the total AM j = l + σ [33]. To illustrate this, the experimental and numerical results of the SPPs for different combinations of SAM and OAM are summarized in Fig. 5 . The NSOM images are shown in the first column with the corresponding simulated transversal component distributions of the SPPs in the second column and longitudinal components distributions of the SPPs in the third column. The number at the right bottom corner and left bottom corner of each image denote the total AM value j and (σ, l) respectively. In Table 2 , the diameter of the center bright ring or the FWHM of the center bright spot are given corresponding to Fig. 5. In the error-allowed range, both of the transversal and the longitudinal SPPs have the same intensity distributions if the excitation photons have the same absolute value of the total AM |j|. There are slight variations in Fig. 5(c2), which is caused by the mode profile differences between fundamental Gaussian beam and beams with OAM = +/−1. The overall feature of the distributions remains the same. These observations indicate that the OAM and the SAM play an equivalent role in effecting the intensity distribution of the SPPs. So a set of OAMs cannot be used to encode information unless no OAM will lead to the same absolute value of the total AM |j|. This property limits the OAM in encoding information but may be useful for encoding information secretly. For instance, one may set SAM as the code key and encode information on the OAM. Since the distributions of the SPPs with the same |j| are indistinguishable, the information cannot be decoded if one doesn’t know the exact value of SAM, which will guarantee the safety of the data transmission. But the information can be successfully decoded by the real receiving end with the correct code key.
Fig. 5 The intensity distributions of the SPPs excited by photons with different total AM. a1, b1, c1 and d1 are the NSOM images of the SPPs with |j| = 3, 2, 1 and 0 respectively. a2, b2, c2 and d2 are the numerical simulations of the transversal SPPs corresponding to a1, b1, c1 and d1. a3, b3, c3 and d3 are the numerical simulations of the longitudinal SPPs with |j| = 3, 2, 1 and 0 respectively. The AM of j and (σ, l) combinations are given at the right bottom corner and left bottom corner of each numerical result, respectively.
Table 2. The Dimeters (D) of the Bright Rings and the FWHMs (F) of the Bright Spots of the SPPs Intensity Distribution Near the Center of the Plasmonic Lens. (Length Unit: Nm)
As shown in Fig. 5, the transversal and longitudinal SPPs distributions are different for the same |j|. On the center of the plasmonic lens, the transversal SPPs interfere constructively to form a bright spot in the case of |j| = 1 and destructively to form a bright ring with a dark center spot in other cases attributed to the phase singularity on the center. For the longitudinal SPPs, however, there is a bright center spot in the case of |j| = 0 and a bright ring with a dark center spot for the other cases. As illustrated in Table 2, the diameters of the bright rings become larger as the |j| increases for both of the transversal and longitudinal SPPs. The difference between the transversal and longitudinal SPPs with the same AM offers two potential schemes for data encoding. That is, the information can be encoded either onto the interference pattern of the longitudinal SPPs or onto the transversal SPPs
4. Conclusions
It is numerically and experimentally proved that both of SAM and OAM information carried by the photons can be transferred to SPPs near field energy distributions via plasmonic nanostructures. The SAM and OAM are independent and play an equivalent role in excitation of the SPPs. The intensity distribution of the SPPs on the metal surface near the center of a nano-ring plasmonic lens depends on the absolute value of the total AM. This work points out a promising means for the information encoding and transmission in nanostructures using SPPs that may find broad applications in the future investigation of integrated plasmonic devices in both classical and quantum information fields.
Acknowledgments
This work was funded by the National Basic Research Program of China (Grants No. 2011CBA00200 and 2011CB921200), the Innovation funds from Chinese Academy of Sciences (Grants No. 60921091), the National Natural Science Foundation of China (Grants No. 10904137 and 10934006), and the Fundamental Research Funds for the Central Universities (Grants No.WK2470000005). X. F. Ren thanks the Program for New Century Excellent Talents in University. G. H. Rui acknowledges the support by the National Basic Research Program of China (Grant No. 2011CB301802).
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